## Reliable Methods for Computer Simulation: Error Control and Posteriori EstimatesRecent decades have seen a very rapid success in developing numerical methods based on explicit control over approximation errors. It may be said that nowadays a new direction is forming in numerical analysis, the main goal of which is to develop methods ofreliable computations. In general, a reliable numerical method must solve two basic problems: (a) generate a sequence of approximations that converges to a solution and (b) verify the accuracy of these approximations. A computer code for such a method must consist of two respective blocks: solver and checker. In this book, we are chiefly concerned with the problem (b) and try to present the main approaches developed for a posteriori error estimation in various problems. The authors try to retain a rigorous mathematical style, however, proofs are constructive whenever possible and additional mathematical knowledge is presented when necessary. The book contains a number of new mathematical results and lists a posteriori error estimation methods that have been developed in the very recent time. · computable bounds of approximation errors · checking algorithms · iteration processes · finite element methods · elliptic type problems · nonlinear variational problems · variational inequalities |

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### Contents

1 | |

7 | |

23 | |

Chapter 4 A posteriori estimates for finite element approximations | 39 |

Chapter 5 Foundations of duality theory | 79 |

Chapter 6 Twosided a posteriori estimates for linear elliptic problems | 125 |

Chapter 7 A posteriori estimates for nonlinear variational problems | 209 |

Chapter 8 A posteriori estimates for variational inequalities | 245 |

281 | |

Notation | 301 |

303 | |

### Common terms and phrases

approximate solution approximation errors arbitrary elements Assume AVuh Banach space boundary conditions boundary-value problem Chapter coincides computed consider continuous function converges convex functionals convex set Definition denote derivatives deviation differential divy dual equations error estimation methods error indication exact solution example Find u e finite element approximations finite element method follows functional G G(Av Galerkin approximation Hilbert space implies the estimate integral integrand linear elasticity Lipschitz continuous lower bound lower semicontinuous mathematical matrix Me(v mesh minimization nonnegative obtain operator piecewise affine post-processing posteriori error estimates posteriori estimates priori Proof properties Proposition quantity residual respective right-hand side saddle point satisfies the condition sequence Sobolev spaces superconvergence supremum term theorem two-sided estimates uniformly convex upper bound variational inequalities variational problems vector Wu e V0 zero

### Popular passages

Page 15 - If a distribution can be identified with a locally integrable function, then it is called regular.

### References to this book

Lectures on Advanced Computational Methods in Mechanics Johannes Kraus,Ulrich Langer Limited preview - 2007 |

Finite Elemente: Theorie, schnelle Löser und Anwendungen in der ... Dietrich Braess No preview available - 2007 |